Evaluate the following integrals: 2x ^4x+4 ^2x-2 dx - Vidyadip

Question

Evaluate the following integrals:

$\int\frac{\sin2\text{x}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}\text{ dx}$

✓

Answer

$\int\frac{\sin(2\text{x})\text{dx}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}$
Let $\sin^2\text{x}=\text{t}$
$\Rightarrow2\sin\text{x}\cos\text{x}\text{ dx}=\text{dt}$
$\Rightarrow\sin(2\text{x})\text{ dx}=\text{dt}$
Now, $\int\frac{\sin(2\text{x})\text{dx}}{\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}}$
$=\int\frac{\text{dt}}{\sqrt{\text{t}^2+4\text{t}-2}}$
$=\int\frac{\text{dt}}{\sqrt{\text{t}^2+4\text{t}+4-4-2}}$
$=\int\frac{\text{dt}}{\sqrt{(\text{t}+2)^2-\big(\sqrt6\big)^2}}$
$=\log\Big|\text{t}+2+\sqrt{(\text{t}+2)^2-6}\Big|+\text{C}$
$=\log\Big|\text{t}+2+\sqrt{\text{t}^2+4\text{t}-2}\Big|+\text{C}$
$=\log\Big|\sin^2\text{x}+2\sqrt{\sin^4\text{x}+4\sin^2\text{x}-2}\Big|+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the value of k in this question, so that the function f is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\frac{\sqrt{1+\text{kx}}-\sqrt{1-\text{kx}}}{\text{x}},&\text{if}-1\leq0\\\frac{2\text{x}+1}{\text{x}-1},&\text{if }0\leq\text{x}\leq1\end{cases}$ at x = 0.

Evaluate the following integrals:
$\int_\limits{\frac{1}{3}}^{1}\frac{\big(\text{x}-\text{x}^3\big)^{\frac{1}{3}}}{\text{x}^4}\text{ dx}$

Calculate the area of the ragion bounded by y²= x and x²= y.

Form the differential equation corresponding to $\text{y}^2-2\text{ay}+\text{x}^2=\text{a}^2$ by eliminating a.

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{5\text{x}}{1+6\text{x}^3}\Big), -\frac{1}{\sqrt{6}}<\text{x}<\frac{1}{\sqrt{6}}$

A card is drawn and replaced in an ordinary pack of 52 cards. How many times must a card be drawn so that.

there is at least an even chance of drawing a heart.
the probability of drawing a heart is greater than $\frac{3}{4}$?

Solve the following differential equation:
$\text{y dx}+\Big\{\text{x}\log\Big(\frac{\text{y}}{\text{x}}\Big)\Big\}\text{dy}-2\text{x dy}=0$

Two tailors A and B earn Rs 150 and Rs 200 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.

If f(x) = x³ + 7x² + 8x - 9, find f(4).

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}1&43&6\\7&35&4\\3&17&2\end{vmatrix}$